标题： 舒尔-萨托关于拟椭圆环的理论 显示英文标题

标题： Schur-Sato theory for quasi-elliptic rings

摘要： 准椭圆环的概念是由于试图对可积系统理论中发现的一类广泛的交换环进行分类而出现的，例如交换微分、差分、微分差分等算子的环。 它们包含在一个特定的非交换“通用”环中——流形上伪微分算子环的纯代数类似物，并且在某些轻微限制下，可以进行方便的代数几何描述。 这个描述的重要代数部分是Schur-Sato理论——对普通微分算子的著名理论的推广。 这个理论的一些部分在一系列论文中较早地被发展，主要是针对二维情况。 在本文中，我们以任意维数呈现这一理论。 我们将这一理论应用于证明两个关于准椭圆环的分类定理，这些定理是通过某些子空间对（Schur对）来表述的。 它们对于上述准椭圆环的代数几何描述是必要的。 该理论是有效的，并且有其他几个应用，其中包括对Abhyankar反演公式的新的证明。 显示英文摘要

摘要： The notion of quasi-elliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference, differential-difference, etc. operators. They are contained in a certain non-commutative "universal" ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic-geometric description. An important algebraic part of this description is the Schur-Sato theory - a generalisation of the well known theory for ordinary differential operators. Some parts of this theory were developed earlier in a series of papers, mostly for dimension two. In this paper we present this theory in arbitrary dimension. We apply this theory to prove two classification theorems of quasi-elliptic rings in terms of certain pairs of subspaces (Schur pairs). They are necessary for the algebraic-geometric description of quasi-elliptic rings mentioned above. The theory is effective and has several other applications, among them is a new proof of the Abhyankar inversion formula.